Homotopy Quantum Field Theory

同伦量子场论（HQFT）是拓扑量子场论的一个分支，研究具有给定目标空间映射的流形和协阵。该项目的目的是在目标空间为K（G,1）型Eilenberg-MacLane空间的情况下建立三维HQFT的基础，其中G是一个离散群（有限或无限）。这应该概括了N. Reshetikhin， O. Viro和A. Virelizier的PI的已知结果。该项目将具体解决以下问题：(1)从球面g -聚变g -类别中生成三维hqft的状态和构造；(2)引入了g模g范畴的适当概念，并给出了基于该范畴的三维HQFT的手术构造。这些问题的解决方案将包括定义所需的g类和相应的hqft类。最后，PI将通过Drinfeld-Joyal-Street类别中心的g版本建立结构(1)和(2)之间的基本关系：通过构造(1)从球形g -聚变g -类别C获得的HQFT等同于通过构造(2)从适当定义的C的g -中心获得的HQFT。该项目开发了由量子物理思想产生的几何新技术。具体来说，该项目将侧重于拓扑学场论的概念，该概念是由菲尔兹奖得主爱德华·威滕在20世纪80年代提出的。拓扑场论允许拓扑学家从微观的角度分析几何物体的形状，将它们分解成小的基本块。PI和合著者给出的拓扑场论的数学公式成功地在以前不相关的纯数学领域之间建立了桥梁，比如一方面是空间中打结弦的几何理论，另一方面是表示和范畴的代数理论。该项目极大地扩大了可以使用拓扑场论分析的几何对象的类别。我们不仅考虑对象本身，而且考虑它们之间的关系，在数学中称为映射或映射。本课题的主要目的是从上述微观的观点来分析这些图，并推导出相应的代数和范畴论中的概念。该项目将介绍和发展两种不同的解决方案来解决这个问题，并建立一个微妙的定理，将这两个解决方案联系起来。这个项目将有助于更好地理解几何物体及其地图。它还将产生新的强大的代数概念和技术。纯数学之外的潜在应用领域包括理论物理和量子计算。

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory dealing with manifolds and cobordisms endowed with maps to a given target space. The aim of the project is to establish foundations of 3-dimensional HQFT in the case where the target space is the Eilenberg-MacLane space of type K(G,1) where G is a discrete group (finite or infinite). This should generalize the known results of the PI with N. Reshetikhin, O. Viro, and A. Virelizier. The project will specifically address the following problems:(1) produce a state sum construction of 3-dimensional HQFTs from spherical G-fusion G-categories; (2) introduce an appropriate notion of a G-modular G-category and give a surgery construction of a 3-dimensional HQFT from such a category. Solutions to these problems will include definitions of the required classes of G-categories and of the corresponding HQFTs. Finally, the PI will establish a fundamental relation between the constructions (1) and (2) via a G-version of the Drinfeld-Joyal-Street center of categories: the HQFT obtained by construction (1) from a spherical G-fusion G-category C is equivalent to the HQFT obtained by construction (2) from the appropriately defined G-center of C. The project develops new techniques in geometry arising from ideas of quantum physics. Specifically, the project will focus on the notion of Topological Field Theory introduced by the Fields medalist Edward Witten in the 1980's. Topological Field Theory allows topologists to analyze the shape of geometric objects from the microscopic viewpoint decomposing them into small elementary pieces. Mathematical formulation of Topological Field Theory given by the PI and co-authors has successfully led to creating bridges between previously unrelated areas of pure mathematics like the geometric theory of knotted strings in space on one hand, and the algebraic theory of representations and categories on the other hand. The project considerably enlarges the class of geometric objects which can be analyzed using Topological Field Theory. We include in consideration not only the objects themselves but also relations between them known in mathematics as maps or mappings. The principal aim of the project is to analyze the maps from the above mentioned microscopic viewpoint and to derive corresponding notions in algebra and theory of categories. The project will introduce and develop two different solutions to this problem and establish a subtle theorem relating these two solutions. This project will lead to a better understanding of geometric objects and their maps. It will also produce new powerful algebraic notions and techniques. Potential areas of applications outside of pure mathematics include theoretical physics and quantum computations.